fixed point theorem in ordered metric spaces for

Nonlinear Functional Analysis and Applications Vol. 17, No. 3 (2012), pp. 301-306 http://nfaa.kyungnam.ac.kr/jour-nfaa.htm c 2012 Kyungnam University Press Copyright

FIXED POINT THEOREM IN ORDERED METRIC SPACES FOR GENERALIZED CONTRACTION MAPPINGS SATISFYING RATIONAL TYPE EXPRESSIONS Sumit Chandok1 and Jong Kyu Kim2 1 Department of Mathematics Khalsa College of Engineering & Technology Punjab Technical University Amritsar-143001, India e-mail: [email protected], [email protected] 2 Department of Mathematics Education, Kyungnam University

Changwon, 631-701, Korea e-mail: [email protected] Abstract. In this paper, we prove a fixed point result in the framework of partially ordered metric spaces satisfying a generalized contractive condition of rational type. The result generalize and extend some known results in the literature.

1. Introduction and Preliminaries In [9], Jaggi and Dass proved the following fixed point theorem. Theorem 1.1. Let T be a continuous self map defined on a complete metric space (X, d). Suppose that T satisfies the following contractive condition: d(x, T x)d(y, T y) d(T x, T y) ≤ α + β(d(x, y)) (1.1) d(x, y) + d(x, T y) + d(y, T x) for all x, y ∈ X, x ≥ y and for some α, β ∈ [0, 1) with α + β < 1, then T has a unique fixed point in X. The aim of this paper is to give a version of Theorem 1.1 in partially ordered metric spaces. 0

Received May 7, 2012. Revised July 5, 2012. 2000 Mathematics Subject Classification: 46T99, 41A50, 47H10, 54H25. 0 Keywords: Fixed point, rational type generalized contraction mappings, ordered metric spaces. 0

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S. Chandok and J.K. Kim

The Banach contraction mapping is one of the pivotal results of analysis. It is very popular tool for solving existence problems in many different fields of mathematics. There are a lot of generalizations of the Banach contraction principle in the literature (see [1]-[11] and references cited therein). Ran and Reurings [11] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and Rod´riguez-L´opez [10] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [2] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results on a first order differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point results in metric spaces and partially ordered metric spaces. The purpose of this paper is to establish a fixed point result satisfying a generalized contraction mappings of rational type in partially ordered metric spaces. 2. Main Results Definition 2.1. Suppose (X, ≤) is a partially ordered set and T : X → X. T is said to be monotone nondecreasing if for all x, y ∈ X, x≤y

implies

T x ≤ T y.

(2.1)

Theorem 2.2. Let (X, ≤) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose that T is a continuous self-mapping on X, T is monotone nondecreasing mapping and d(x, T x)d(y, T y) + β(d(x, y)) (2.2) d(T x, T y) ≤ α d(x, y) + d(x, T y) + d(y, T x) for all x, y ∈ X, x ≥ y and for some α, β ∈ [0, 1) with α + β < 1. If there exists x0 ∈ X with x0 ≤ T x0 , then T has a fixed point. Proof. If T x0 = x0 , then we have the result. Suppose that x0 < T x0 . Since T is a monotone nondecreasing mapping, we obtain by induction that x0 < T x0 ≤ T 2 x0 ≤ . . . ≤ T n x0 ≤ T n+1 x0 ≤ . . . .

(2.3)

By induction, we can construct a sequence {xn } in X such that xn+1 = T xn , for every n ≥ 0. Since T is monotone nondecreasing mapping, we obtain x0 ≤ x1 ≤ x2 ≤ . . . ≤ xn ≤ xn+1 ≤ . . . . If there exists n ≥ 1 such that xn+1 = xn , then from xn+1 = T xn = xn , xn is a fixed point and the proof is finished. Suppose that xn+1 6= xn , for all n ≥ 1.

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Since xn > xn−1 , for all n ≥ 1, from (2.2), we have d(xn+2 , xn+1 ) = d(T xn+1 , T xn ) d(xn+1 , T xn+1 )d(xn , T xn ) ≤ α d(xn+1 , xn ) + d(xn+1 , T xn ) + d(xn , T xn+1 ) +β(d(xn+1 , xn )) d(xn+1 , xn+2 )d(xn , xn+1 ) = α + β(d(xn+1 , xn )) d(xn+1 , xn ) + d(xn , xn+2 ) d(xn+1 , xn+2 )d(xn , xn+1 ) ≤ α + β(d(xn+1 , xn )) d(xn+1 , xn+2 ) = α(d(xn , xn+1 )) + β(d(xn+1 , xn )) = (α + β)d(xn+1 , xn ), (2.4) which implies that d(xn+2 , xn+1 ) ≤ (α + β)d(xn+1 , xn ).

(2.5)

Using, mathematical induction we have d(xn+2 , xn+1 ) ≤ (α + β)n+1 d(x1 , x0 ).

(2.6)

Put k = α + β < 1. Now, we shall prove that {xn } is a Cauchy sequence. For m ≥ n, we have d(xm , xn ) ≤ d(xm , xm−1 ) + d(xm−1 , xm−2 ) + ... + d(xn+1 , xn ) ≤ k m−1 + k m−2 + ... + k n d(x1 , x0 ) n k ≤ d(x1 , x0 ), (2.7) 1−k which implies that d(xm , xn ) → 0, as m, n → ∞. Thus {xn } is a Cauchy sequence in a complete metric space X. Therefore, there exits u ∈ X such that limn→∞ xn = u. By the continuity of T , we have T u = T ( lim xn ) = lim T xn = lim xn+1 = u. n→∞

Hence u is a fixed point of T .

n→∞

n→∞

In what follows, we prove that Theorem 2.2 is still valid for T , not necessarily continuous, assuming the following hypothesis in X. If {xn } is a non-decreasing sequence in X such that xn → x, then x = sup{xn }. Theorem 2.3. Let (X, ≤) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose

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that T is a self-mapping on X, T is monotone nondecreasing mapping and d(x, T x)d(y, T y) + β(d(x, y)) (2.8) d(T x, T y) ≤ α d(x, y) + d(x, T y) + d(y, T x) for all x, y ∈ X, x ≥ y and for some α, β ∈ [0, 1) with α + β < 1. Assume that {xn } is a non-decreasing sequence in X such that xn → x, then x = sup{xn }. If there exists x0 ∈ X with x0 ≤ T x0 , then T has a fixed point. Proof. Following the proof of Theorem 2.2, we have {xn } is a Cauchy sequence. Since {xn } is a non-decreasing sequence in X such that xn → u, then u = sup{xn }. Particularly, xn ≤ u for all n ∈ N. Since T is monotone nondecreasing mapping T xn ≤ T u, for all n ∈ N or, equivalently, xn+1 ≤ T u, for all n ∈ N. Moreover, as xn < xn+1 ≤ T u and u = sup{xn }, we get u ≤ T u. Construct a sequence {yn } as y0 = u, yn+1 = T yn , for all n ≥ 0. Since y0 ≤ T y0 , arguing like above part, we obtain that {yn } is a non-decreasing sequence and limn→∞ yn = y for certain y ∈ X, so we have y = sup{yn }. Since xn < u = y0 ≤ T u = T y0 ≤ yn ≤ y, for all n, using (2.8), we have d(xn+1 , yn+1 ) = d(T xn , T yn ) d(xn , T xn )d(yn , T yn ) ≤ α d(xn , yn ) + d(xn , T yn ) + d(yn , T xn ) +β(d(xn , yn )) d(xn , xn+1 )d(yn , yn+1 ) = α d(xn , yn ) + d(xn , yn+1 ) + d(yn , xn+1 ) +β(d(xn , yn )).

(2.9)

Letting n → ∞, we have d(u, y) ≤ βd(u, y). As β < 1, we have d(u, y) = 0. Particularly, u = y = sup{yn }, and consequently, u ≤ T u ≤ u. Hence we conclude that u is a fixed point of T . Now, we shall prove the uniqueness of the fixed point. Theorem 2.4. In addition to the hypotheses of Theorem 2.2 (or Theorem 2.3), suppose that for every x, y ∈ X, there exists z ∈ X that is comparable to x and y, then T has a unique fixed point. Proof. From Theorem 2.2 (or Theorem 2.3), the set of fixed points of T is non-empty. Suppose that x, y ∈ X are two fixed points of T . We distinguish two cases:

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Case 1. If x and y are comparable and x 6= y, then using (2.2), we have d(x, y) = d(T x, T y) d(x, T x)d(y, T y) ≤ α + β(d(x, y)) d(x, y) + d(x, T y) + d(y, T x) = β(d(x, y)), which implies that d(x, y) = 0, as β < 1. Hence x = y. Case 2. If x is not comparable to y, there exists z ∈ X that is comparable to x and y. Monotonicity implies that that T n z is comparable to T n x = x and T n y = y for n = 0, 1, 2, . . .. If there exists n0 ≥ 1 such that T n0 z = x, then as x is a fixed point, the sequence {T n z : n ≥ n0 } is constant, and, consequently, limn→∞ T n z = x. On the other hand, if T n z 6= x for n ≥ 1, using the contractive condition, we obtain, for n ≥ 2, d(T n z, x) = d(T n z, T n x) d(T n−1 x, T n x)d(T n−1 z, T n z) ≤ α d(T n−1 x, T n−1 z) + d(T n−1 x, T n z) + d(T n−1 z, T n x) +β(d(T n−1 x, T n−1 z)) d(x, x)d(T n−1 z, T n z) = α + β(d(x, T n−1 z)) d(x, T n−1 z) + d(x, T n z) + d(T n−1 z, x) = β(d(x, T n−1 z)), which implies that d(T n z, x) ≤ β(d(T n−1 z, x)). Using mathematical induction, we have d(T n z, x) ≤ β n (d(z, x)), for n ≥ 2, and as β < 1, we have limn→∞ T n z = x. Using a similar argument, we can prove that limn→∞ T n z = y. Now, the uniqueness of the limit implies x = y. Hence T has a unique fixed point. Other consequences of our results are the following for the mappings involving contractions of integral type. Denote by Λ the set of functions µ : [0, ∞) → [0, ∞) satisfying the following hypotheses: (h1) µ is a Lebesgue-integrable R mapping on each compact subset of [0, ∞); (h2) for any > 0, we have 0 µ(t)dt > 0. Corollary 2.5. Let (X, ≤) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose that T is a continuous self-mapping on X, T is monotone nondecreasing mapping and d(x,T x)d(y,T y) Z d(T x,T y) Z d(x,y) Z d(x,y)+d(y,T x)+d(x,T y) ψ(t)dt ≤ α ψ(t)dt + β ψ(t)dt 0

0

0


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for all x, y ∈ X for which x and y are comparable, ψ ∈ Λ and for some α, β ∈ [0, 1) with α + β < 1. If there exists x0 ∈ X with x0 ≤ T x0 , then T has a fixed point. Corollary 2.6. Let (X, ≤) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose that T is a self-mapping on X, T is monotone nondecreasing mapping and d(x,T x)d(y,T y) Z d(x,y) Z Z d(T x,T y) d(x,y)+d(x,T y)+d(y,T x) ψ(t)dt, ψ(t)dt + β ψ(t)dt ≤ α 0

0

0

for all x, y ∈ X for which x and y are comparable, ψ ∈ Λ and for some α, β ∈ [0, 1) with α + β < 1. Assume that {xn } is a non-decreasing sequence in X such that xn → x, then x = sup{xn }. If there exists x0 ∈ X with x0 ≤ T x0 , then T has a fixed point. References [1] R. P. Agarwal, M. A. El-Gebeily and D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008), 109–116. [2] T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379–1393. [3] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 29 (2002), 531–536. [4] S. Chandok, Some common fixed point theorems for generalized f -weakly contractive mappings, J. Appl. Math. Informatics 29 (2011), 257-265. [5] S. Chandok, Some common fixed point theorems for generalized nonlinear contractive mappings, Comput. Math. Appl. 62 (2011), 3692-3699 DOI: 10.1016/j.camwa.2011.09.009. [6] S. Chandok, Common fixed points for generalized nonlinear contractive mappings in metric spaces, To appear in Mat. Vesnik. [7] J. Harjani, B. Lopez and K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Abstract Appl. Anal. 2010, Article ID 190701. [8] D. S. Jaggi, Some unique fixed point theorems, Indian J. Pure Appl. Math. 8 (1977), 223-230. [9] D. S. Jaggi and B. K. Das, An extension of Banachs fixed point theorem through rational expression, Bull. Cal. Math. Soc. 72 (1980), 261-264. [10] J. J. Nieto and R. R. L´ opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239. [11] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132(5) (2004), 1435-1443.